On the selection of a better radial basis function and its shape parameter in interpolation problems

A traditional criterion to calculate the numerical stability of the interpolation matrix is its standard condition number. In this paper, it is observed that the effective condition number (κ_eff) is more informative than the standard condition number (κ) in investigating the numerical stability of the interpolation problem. While the κ_eff considers the function to be interpolated, the standard condition number only depends on the interpolation matrix. We propose using the shape parameter corresponding to the maximum κ_eff to obtain a small error in RBF interpolation. It is also observed that the κ_eff helps to predict the error behavior with respect to the type of the RBF, where the basis function with a higher effective condition number yields a smaller error. In the end, we conclude that the effective condition number links to the error with respect to the selection of a radial basis function, choosing its shape parameter, number of collocation points, and test function. To this end, ten test functions are interpolated using the multiquadric, Matern family, and Gaussian basis functions to show the advantage of the proposed method.